Question

Given v = 2(ab + bc + ca), how do you solve for a?

Answer 1

Hint: To get the right answer we are going to first multiply 2 to the terms in the bracket, after that we will take the terms present with ‘a’ to the side of equal sign and keep the rest to the left side of the equal sign. Finally, we will divide the whole equation with the coefficient of ‘a’ and get the required answer.

Complete step-by-step answer:
We will consider the given equation v = 2(ab + bc + ca) and multiply 2 by 2(ab + bc + ca). This will result in the equation v = 2ab + 2bc + 2ca. Now, we will take 2bc and shift it to the left hand side of the equation. Thus, we get v – 2bc = 2ab + 2 ac.
As we can clearly see that ‘a’ can be taken out as a common variable, so we get v – 2bc = a(2b + 2c). The resulting equation must be divided by (2b + 2c). Thus, we get $a=\dfrac{v-2bc}{2b+2c}$.
Hence, the desired value of $a=\dfrac{v-2bc}{2b+2c}$.

Note: In this question, we have taken ‘a’ as a common variable because we need to find the value of ‘a’. But in case we were supposed to find the value of b or c then we would have taken b or c as a common variable and lead to the answer. It must be noted that the terms while arranging to the either side of the equation needs to get a sign changed. An alternative approach to this question is to apply cross multiplication while getting to v – 2bc = a(2b + 2c). equation. This can be done as,
$\begin{align}
  & \dfrac{v-2bc}{1}=\dfrac{a\left( 2b+2c \right)}{1} \\
 & \Rightarrow a=\dfrac{v-2bc}{2b+2c} \\
\end{align}$
Do focus on the question while solving it as there might be a chance of leading towards the wrong answer which is not required.